Explicit deformation of Galois representations.
Explicit descent for Jacobians of cyclic coevers of the projective line.
Explicit determination of the images of the Galois representations attached to abelian surfaces with .
Explicit digital inversive pseudorandom numbers
Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)
We prove that the error term differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.
Explicit Estimates in the Arithmetic Theory of Cusp Forms and Poincaré Series.
Explicit estimates on the summatory functions of the Möbius function with coprimality restrictions
We prove that for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,
Explicit evaluation of certain exponential sums.
Explicit evaluations of quadruple Euler sums
Explicit evaluations of some Weil sums
Explicit evaluations of the Rogers-Ramanujan continued fraction.
Explicit form for the discrete logarithm over the field
For generator of the multiplicative group of the field , the discrete logarithm of an element of the field to the base , is that integer , . The -ary digits which represent can be described with extremely simple polynomial forms.
Explicit form of the modular equation.
Explicit forms of Kummer's complementary theorems to his law of quintic reciprocity.
Explicit formulae of Siegel Eisenstein series.
Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions.
Explicit formulas for Bernoulli and Euler numbers.
Explicit formulas for strong Davenport pairs
Explicit formulas for the constituent matrices. Application to the matrix functions
We present a constructive procedure for establishing explicit formulas of the constituents matrices. Our approach is based on the tools and techniques from the theory of generalized Fibonacci sequences. Some connections with other results are supplied. Furthermore,we manage to provide tractable expressions for the matrix functions, and for illustration purposes we establish compact formulas for both the matrix logarithm and the matrix pth root. Some examples are also provided.
Explicit formulas for the eigenvalues of Hecke operators